Properties

 Label 1856.4.a.l Level $1856$ Weight $4$ Character orbit 1856.a Self dual yes Analytic conductor $109.508$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1856 = 2^{6} \cdot 29$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1856.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$109.507544971$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{6})$$ Defining polynomial: $$x^{2} - 6$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 58) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta ) q^{3} + ( 5 - 6 \beta ) q^{5} + ( -8 - 8 \beta ) q^{7} + ( -20 + 2 \beta ) q^{9} +O(q^{10})$$ $$q + ( 1 + \beta ) q^{3} + ( 5 - 6 \beta ) q^{5} + ( -8 - 8 \beta ) q^{7} + ( -20 + 2 \beta ) q^{9} + ( 45 + 3 \beta ) q^{11} + ( 25 - 8 \beta ) q^{13} + ( -31 - \beta ) q^{15} + ( -22 + 16 \beta ) q^{17} + ( -54 + 4 \beta ) q^{19} + ( -56 - 16 \beta ) q^{21} + ( -14 - 78 \beta ) q^{23} + ( 116 - 60 \beta ) q^{25} + ( -35 - 45 \beta ) q^{27} -29 q^{29} + ( 33 + 109 \beta ) q^{31} + ( 63 + 48 \beta ) q^{33} + ( 248 + 8 \beta ) q^{35} + ( -20 - 4 \beta ) q^{37} + ( -23 + 17 \beta ) q^{39} + ( 152 - 80 \beta ) q^{41} + ( 65 - 53 \beta ) q^{43} + ( -172 + 130 \beta ) q^{45} + ( -257 - 99 \beta ) q^{47} + ( 105 + 128 \beta ) q^{49} + ( 74 - 6 \beta ) q^{51} + ( 479 - 52 \beta ) q^{53} + ( 117 - 255 \beta ) q^{55} + ( -30 - 50 \beta ) q^{57} + ( 90 - 250 \beta ) q^{59} + ( -514 + 12 \beta ) q^{61} + ( 64 + 144 \beta ) q^{63} + ( 413 - 190 \beta ) q^{65} + ( 456 + 20 \beta ) q^{67} + ( -482 - 92 \beta ) q^{69} + ( 398 + 34 \beta ) q^{71} + ( -428 + 176 \beta ) q^{73} + ( -244 + 56 \beta ) q^{75} + ( -504 - 384 \beta ) q^{77} + ( -159 - 361 \beta ) q^{79} + ( 235 - 134 \beta ) q^{81} + ( 914 + 38 \beta ) q^{83} + ( -686 + 212 \beta ) q^{85} + ( -29 - 29 \beta ) q^{87} + ( -472 + 72 \beta ) q^{89} + ( 184 - 136 \beta ) q^{91} + ( 687 + 142 \beta ) q^{93} + ( -414 + 344 \beta ) q^{95} + ( 184 + 100 \beta ) q^{97} + ( -864 + 30 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 10 q^{5} - 16 q^{7} - 40 q^{9} + O(q^{10})$$ $$2 q + 2 q^{3} + 10 q^{5} - 16 q^{7} - 40 q^{9} + 90 q^{11} + 50 q^{13} - 62 q^{15} - 44 q^{17} - 108 q^{19} - 112 q^{21} - 28 q^{23} + 232 q^{25} - 70 q^{27} - 58 q^{29} + 66 q^{31} + 126 q^{33} + 496 q^{35} - 40 q^{37} - 46 q^{39} + 304 q^{41} + 130 q^{43} - 344 q^{45} - 514 q^{47} + 210 q^{49} + 148 q^{51} + 958 q^{53} + 234 q^{55} - 60 q^{57} + 180 q^{59} - 1028 q^{61} + 128 q^{63} + 826 q^{65} + 912 q^{67} - 964 q^{69} + 796 q^{71} - 856 q^{73} - 488 q^{75} - 1008 q^{77} - 318 q^{79} + 470 q^{81} + 1828 q^{83} - 1372 q^{85} - 58 q^{87} - 944 q^{89} + 368 q^{91} + 1374 q^{93} - 828 q^{95} + 368 q^{97} - 1728 q^{99} + O(q^{100})$$

Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.44949 2.44949
0 −1.44949 0 19.6969 0 11.5959 0 −24.8990 0
1.2 0 3.44949 0 −9.69694 0 −27.5959 0 −15.1010 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$29$$ $$1$$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1856.4.a.l 2
4.b odd 2 1 1856.4.a.i 2
8.b even 2 1 58.4.a.c 2
8.d odd 2 1 464.4.a.e 2
24.h odd 2 1 522.4.a.j 2
40.f even 2 1 1450.4.a.g 2
232.g even 2 1 1682.4.a.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
58.4.a.c 2 8.b even 2 1
464.4.a.e 2 8.d odd 2 1
522.4.a.j 2 24.h odd 2 1
1450.4.a.g 2 40.f even 2 1
1682.4.a.c 2 232.g even 2 1
1856.4.a.i 2 4.b odd 2 1
1856.4.a.l 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1856))$$:

 $$T_{3}^{2} - 2 T_{3} - 5$$ $$T_{5}^{2} - 10 T_{5} - 191$$ $$T_{7}^{2} + 16 T_{7} - 320$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-5 - 2 T + T^{2}$$
$5$ $$-191 - 10 T + T^{2}$$
$7$ $$-320 + 16 T + T^{2}$$
$11$ $$1971 - 90 T + T^{2}$$
$13$ $$241 - 50 T + T^{2}$$
$17$ $$-1052 + 44 T + T^{2}$$
$19$ $$2820 + 108 T + T^{2}$$
$23$ $$-36308 + 28 T + T^{2}$$
$29$ $$( 29 + T )^{2}$$
$31$ $$-70197 - 66 T + T^{2}$$
$37$ $$304 + 40 T + T^{2}$$
$41$ $$-15296 - 304 T + T^{2}$$
$43$ $$-12629 - 130 T + T^{2}$$
$47$ $$7243 + 514 T + T^{2}$$
$53$ $$213217 - 958 T + T^{2}$$
$59$ $$-366900 - 180 T + T^{2}$$
$61$ $$263332 + 1028 T + T^{2}$$
$67$ $$205536 - 912 T + T^{2}$$
$71$ $$151468 - 796 T + T^{2}$$
$73$ $$-2672 + 856 T + T^{2}$$
$79$ $$-756645 + 318 T + T^{2}$$
$83$ $$826732 - 1828 T + T^{2}$$
$89$ $$191680 + 944 T + T^{2}$$
$97$ $$-26144 - 368 T + T^{2}$$